Updated at 5:05 p.m. ET: Opposition protesters clashed with police in several Egyptian cities Friday after new Islamist President Mohammed Morsi awarded himself sweeping new powers.

Police fired tear gas in an attempt to disperse tens of thousands of protesters in Cairo’s Tahrir Square, the center of anti-regime protests that ousted longtime U.S.-backed leader Hosni Mubarak in 2011.

"The people want to bring down the regime," shouted protesters, echoing a chant used in the anti-Mubarak uprising. "Get out, Morsi," they chanted.

State TV also said Morsi opponents set fire to Muslim Brotherhood offices in the Suez Canal cities of Suez, Port Said and Ismailia.

Clashes also erupted between police and opposition protesters in the Mediterranean city of Alexandria, the southern city of Assiut and in Giza, the sister city of the capital. In Alexandria, Morsi opponents hurled stones at Brotherhood supporters outside a mosque and stormed a nearby office of the group.

However, Muslim Brotherhood backers gathered in front of the presidential palace in northern Cairo to support Morsi -- illustrating a widening gulf over Egypt’s future.

Tarek Fawzy / AP

Protesters opposed to President Mohammed Morsi break into the offices of the Freedom and Justice Party, the political wing of the Muslim Brotherhood, in Alexandria, Egypt, on Friday.

Wide powersBuoyed by accolades from around the world for mediating a truce between Hamas and Israel, Morsi on Thursday ordered that an Islamist-dominated assembly writing the new constitution could not be dissolved by legal challenges.

Other changes give Morsi power to take security measures to protect his position, which rights groups say are like new emergency laws.

"I am for all Egyptians. I will not be biased against any son of Egypt,'' Morsi told the crowd outside the presidential palace, adding that he was working for social and economic stability and the rotation of power.

"Opposition in Egypt does not worry me, but it has to be real and strong,'' he said in response to his critics.

Morsi also said Friday that his government would pay $5,000 to the families of those who died in the protests to oust Mubarak and $3,333 to those who were injured.

The U.S. State Department signaled its concern Friday over Morsi’s declarations.

"One of the aspirations of the revolution was to ensure that power would not be overly concentrated in the hands of any one person or institution," said spokeswoman Victoria Nuland. "The current constitutional vacuum in Egypt can only be resolved by the adoption of a constitution that includes checks and balances, and respects fundamental freedoms, individual rights, and the rule of law consistent with Egypt's international commitments."

Nuland called for calm and for all parties in Egypt to resolve differences through "democratic dialogue."

Meanwhile, the United Nations expressed serious concerns Friday about human rights and stability in Egypt.

Tarek Fawzy / AP

Protesters hurl stones during clashes between supporters and opponents of President Mohammed Morsi in Alexandria, Egypt, on Friday.

"We are very concerned about the possible huge ramifications of this declaration on human rights and the rule of law in Egypt," Rupert Colville, U.N. Human Rights Commissioner Navi Pillay's spokesman, told a news briefing at the United Nations in Geneva. "We also fear this could lead to a very volatile situation over the next few days, starting today in fact."

Morsi's decree is also bound to worry Western allies, particularly the United States, a generous benefactor to Egypt's army.

NBC News' Charlene Gubash, Reuters and The Associated Press contributed to this report.